Resilience of spatial networks with inter-links behaving as an external field

TL;DR

This paper models the resilience of spatially embedded interlinked networks using percolation theory, revealing how inter-links act like an external field and analyzing their critical behavior and robustness.

Contribution

It introduces a novel framework applying percolation theory to interlinked spatial networks, characterizing inter-links as an external field and deriving critical exponents.

Findings

Inter-links behave as an external field near percolation transition.

Critical exponents follow Widom's scaling relations.

Framework helps optimize real-world infrastructure resilience.

Abstract

Many real systems such as, roads, shipping routes, and infrastructure systems can be modeled based on spatially embedded networks. The inter-links between two distant spatial networks, such as those formed by transcontinental airline flights, play a crucial role in optimizing communication and transportation over such long distances. Still, little is known about how inter-links affect the resilience of such systems. Here, we develop a framework to study the resilience of interlinked spatially embedded networks based on percolation theory. We find that the inter-links can be regarded as an external field near the percolation phase transition, analogous to a magnetic field in a ferromagnetic-paramagnetic spin system. By defining the analogous critical exponents $\delta$ and $\gamma$, we find that their values for various inter-links structures follow Widom's scaling relations. Furthermore, we study the optimal robustness of our model and compare it with the analysis of real-world networks. The framework presented here not only facilitates the understanding of phase transitions with external fields in complex networks but also provides insight into optimizing real-world infrastructure networks and a magnetic field in a management.